Introducing relations between activities and goods consumption in

microeconomic time use models

Sergio R. Jara-Díaz *

Universidad de Chile

Transport System Division

Casilla 228-3, Santiago, Chile.

Phone: (562) 29784380, Fax: (562)6894206

Email: jaradiaz@ing.uchile.cl

Sebastian Astroza

The University of Texas at Austin

Department of Civil, Architectural and Environmental Engineering

301 E. Dean Keeton St. Stop C1761, Austin TX 78712-1172

Phone: 512-471-4535, Fax: 512-475-8744

Email: sastroza@utexas.edu

Chandra R. Bhat

The University of Texas at Austin

Department of Civil, Architectural and Environmental Engineering

1 University Station C1761, Austin, TX 78712-0278

Phone: 512-471-4535, Fax: 512-475-8744

Email: bhat@mail.utexas.edu

Marisol Castro

Universidad de Chile

Transport System Division

Casilla 228-3, Santiago, Chile.

Phone: (562) 29784380, Fax: (562)6894206

Email: marisolcastro@gmail.com

*corresponding author

2

ABSTRACT

We present a microeconomic model for time use and consumption for workers with an improved

treatment of the (technical) relations between goods and time. In addition to the traditional time

and income constraints, an improved set of restrictions involving explicit relations between

consumption of goods and time assigned to activities is included in two versions. In each

version, a system of equations involving a subset of the consumer’s decision variables is

obtained, including (1) work time, (2) activities that are assigned more time than the minimum,

and (3) goods that are consumed above the minimum. The system cannot be solved explicitly in

the endogenous decision variables but is used to set a stochastic system for econometric

estimation through maximum likelihood. The models are applied to analyze weekly time use and

consumption data from Netherlands for year 2012. The results obtained by this new “goods and

time” framework are compared with previous research in terms of the value of leisure and the

value of work, showing substantial differences in the valuation of time.

Keywords: time use model, value of time, leisure, work, microeconomics, time management,

utility theory, utility maximization.

3

1. INTRODUCTION

Workers’ behavior in terms of their use of time has been studied from many perspectives and in

many disciplines, including labor economics and transportation. Among such studies, the

common thread has been the attempt to explain workers’ time use as a function of exogenous

variables with the aim to understand the frequency, duration, and sequence of activity

participations (see Bhat and Koppelman, 1993). A key component of such a time-use analysis is

an understanding of workers’ willingness to pay to decrease travel time, which incorporates

several effects, including the value of doing something else (leisure or work). In particular,

changes in transportation affect travel time and, therefore, have an impact on the allocation of

time to non-travel activities.

Many approaches have been used to understand the allocation and valuation of time. One

of the most popular approaches is the expansion of the basic microeconomic consumer theory by

including time in a utility function that represents unconstrained ordinal preferences and adding

temporal restrictions besides the budget constraint. As known, consumer theory looks at the

individual as if he or she chooses what he or she prefers; from this viewpoint, utility (an

unobservable artifact) is only a construction from which (observable) demand functions can be

obtained. The essence of these models is that the individual assigns money to buy goods and

invests time to undertake activities through a strategic underlying equilibrium mechanism

between money and time; as known, time cannot be “saved” but it can certainly be reallocated

after changes in exogenous conditions (e.g. income, prices). Since these microeconomic models

simultaneously consider time and income constraints and choices involving money and time,

different types of time values can be developed, including value of time as a resource, value of

working time, and value of assigning time to an activity. These values are important in the

evaluation of transportation policies, because the benefits of the reduction of travel time can be

economically measured using the different estimated values of time.

Becker’s study (1965) appears to be the first to include time and its value in

microeconomic consumer theory. Becker proposed final goods—combinations of market goods

and preparation time—as the argument of the utility function and the inclusion of a total time

constraint, time equivalent of the typical total income restriction (see Pollack, 2003 and

Cherchye et al., 2015 for discussions). According to Becker´s framework, the value of time as a

resource is equal to the individual’s wage rate. Some years later, DeSerpa (1971) modified

4

Becker’s model by including directly goods consumption and time allocation in the utility

function. DeSerpa also added technological restrictions, linking the consumption of goods with a

certain minimum time of consumption. DeSerpa was the first to clearly define leisure activities

(those the individual assigns more time than the minimum) and its value, obtaining relations

between the different values of time. As a derivation of the first order conditions, DeSerpa

indicated that the value of leisure is equal to the total value of work (wage rate plus the intrinsic

value of working time); he further indicated that the willingness to pay to save time in an activity

is equal to the value of leisure (the value of doing something else) plus the value of time assigned

to that activity. Evans (1972) proposed a utility function depending only on time assigned to

activities and a new type of restriction linking time assigned to different activities, i.e., the time

assigned to a particular activity could be directly related to the time assigned to another one. As

noted by Jara-Díaz (2003), the money budget constraint in Evans’ model contains a

transformation of activities into the consumption of goods that can be interpreted as another type

of technical relation between goods and time.

Since the theoretical frameworks of Becker, DeSerpa, and Evans, the literature of

microeconomic time use models has expanded in several directions (for a detailed review, see

Jara-Díaz, 2007), including the study of travel time and mode choice within the goods-leisure

tradeoff framework (Train and McFadden, 1978), investigations related to home-production

(Gronau, 1986), time-specific analysis (Pawlak, 2015, López-Ospina et al., 2015) and of course

more theoretical developments regarding the type of restrictions and variables that should be

considered in the consumer theory framework. Thus, building from DeSerpa (1971) and Evans

(1972), Jara-Díaz (2003) showed that there are two types of technical relations between goods

consumed and time assigned to activities. Simply put, they can be stated as minimum activity

times that depend on the amount of goods needed to perform them (a generalization of DeSerpa)

and minimum consumption of goods induced by the activities undertaken (a generalization of

Evans). These two families of relations can be treated as yet additional constraints in a consumer

behavior microeconomic framework including time use, such that exogenous changes (e.g. re-

design of the transit system or improvements in communication systems) will affect these

relations and induce a change in time use patterns.

If a good is consumed, there may be a minimum consumption level or expenditure

associated with that good. Similarly, if an activity type is participated in, there may be a

5

minimum level of time investment required in the participation (for example, taking a child to

the doctor’s office entails some minimum level of time spent at the doctor’s office). Individuals

may generally prefer to strictly stick to the minimum consumption (or expenditure) level for

some goods (let this set of goods be denoted by RG ) , while may consume (or expend) more

than the minimum for some other goods (let this set of goods be denoted by FG ). In a similar

vein, individuals may invest the minimum possible time for certain activity types (let this set of

activity types be RA ), while they may invest more than the minimum for certain activity types

(let this set of activity types be FA , the leisure activities according to DeSerpa).

In their simplest form, both types of technical relations were introduced by Jara-Díaz and

Guevara (2003) and expanded in Jara-Díaz et al. (2008) as exogenously given minimum levels

of good consumption and time allocation, a very simplified manner to account for these types of

constraints. Jara-Díaz et al.’s (2008) formulation considered, as usual, that consumption of

different goods and time assignment to different types of activities are the consumer’s decision

variables. Although quite limited as a representation of the technical constraints, the simple

formulation allowed for a closed analytical solution in three types of variables: (1) time assigned

to activities beyond the minimum (those in FA ), (2) work time, and (3) amount of goods

consumed above the corresponding minimum (those in FG ). By considering additive

interdependent errors in the resulting equation system, the utility parameters can be estimated

and, for the first time, the (marginal) values of leisure and work were actually estimated and

computed. Here, the value of leisure is equal to the value of time as a resource.

However, there is a component of the total value of leisure that is different from the value

of time as a resource. This difference cannot be revealed with Jara-Díaz’s (2008) model because,

as suggested by Konduri et al. (2011) and shown by Jara-Díaz and Astroza (2013), explicit

relations between goods consumed and time assigned are needed. To begin accounting for this,

here we consider two models: one where time allocated to activities impose minimum

consumption of certain goods, a generalization of Evans (1972); and another where goods

consumed impose a minimum necessary time to activities, a generalization of DeSerpa (1971).

Unlike previous empirical models, all these minima become endogenous. That is, we explicitly

tie goods consumption (or expenditures) levels to time-use. Although closed solutions cannot be

obtained in either case, we show that stochastic specifications can be formulated and estimated in

6

both cases. Unfortunately, this cannot be done when both type of constraints are simultaneously

introduced.

To our knowledge, this is the first time such a set of relationships is included in the time

use model formulation. Indeed, while there have been important recent developments in time-use

modeling (including Bhat’s (2008) multiple discrete-continuous choice model, Jara-Díaz et al.’s

(2008) micro-economic model, and the use of a structural equations model by Konduri et al.,

2011 and Dane et al., 2014), all of these efforts recognize that a better treatment of the

(technical) relations between goods consumed and time use is a critical need.

The proposed model is applied to weekly time use and consumption data obtained from

the 2012 LISS (Longitudinal Internet Studies for the Social Sciences) panel. This panel is

administered by CentERdata (www.lissdata.nl) and is representative of the Dutch population.

The LISS panel is a standard social survey, to which a questionnaire was added to gather

information about time use and consumption (Cherchye et al., 2012). Obtaining data on both

time use and goods consumption from the same source is not common and previous works have

needed to develop a methodology to merge time use surveys and consumer expenditure data

(see, for example, the imputation of income and expenses performed by Olguín, 2008, and the

merging of the 2008 American Time Use Survey and the 2008 Consumer Expenditure Survey by

Konduri et al., 2011). To our knowledge, the LISS panel is one of the few surveys in the world

that captures both time allocation and goods consumption information. Previous studies (Colella

and van Soesty, 2013; Rubin, 2015) have used the LISS data to explore the association between

time use, time constraints and consumption, but this is the first study that uses the data to

understand the link between these variables.

The remainder of the paper is structured as follows. In the next section we formulate the

two versions of the microeconomic model. Section 3 contains the stochastic counterpart and

presents the maximum likelihood estimation procedure. Section 4 describes the data, while

Section 5 discusses the empirical results. The final section summarizes the approach and results,

and identifies future research directions.

2. MODEL FORMULATION

2.1 The common elements

Consider the following time use – goods consumption model for workers:

7

j

j

i

iwjiw XTTUMax

),( XT (1)

j

fjjw cXPIwTts )(0.. (2)

)(0 i

iw TT (3)

In equation (1), U is a Cobb-Douglas utility function that depends on time allocation )(T

and good consumption )(X . The time allocation vector T includes the time assigned to work wT

and the time iT assigned to each non-work activity i during time period. The good consumption

vector X contains the consumption level jX (j=1,2,…,J) for each good j, consumed during the

same time period. The parameters of the utility function are a positive constant , the time

parameters w and i for all i, and the consumption parameters j for all j. Note that i , w

and j represent the elasticity of the utility with respect to time assigned to activity i, time

assigned to work, and consumption of good j respectively. These elasticities measure the

responsiveness of utility to a marginal change in levels of good consumption or time assigned to

activities (ceteris paribus). For example, if w = 0.20, a 1% increase in working time would lead

to a 0.20% increase in utility.

The first constraint (Equation 2) is the income constraint that accounts for all expenses

and all types of income. w is the wage rate, I is the income obtained from non-work activities

(such as pensions, gifts and investment returns), jP is the unitary price of good j and fc

represents the total fixed expenditures (those that do not depend on the goods or services

purchased in the period). The second constraint (Equation 3) is the total time constraint for

activity times. The Lagrange multipliers and represent the marginal utility of increasing

available money and increasing available time, respectively.

The novelty in this paper is the family of technological constraints. In addition to the

income constraint and the total time constraint, we include constraints that impose minimum

consumption of goods and minimum allocation of time. We propose two different versions of

our model: a) one model with exogenous minimum time allocations and endogenous minimum

good consumptions (generalizing Evans, 1972), and b) another model with endogenous

8

minimum time allocations (generalizing DeSerpa, 1971) and exogenous minimum good

consumptions.

2.2 Model with endogenous minimum consumption of goods and exogenous minimum time

allocations

In addition to (1)-(3) we propose the inclusion of the following family of constraints:

)(0min

iii iTT (4)

)(0 j

i

iijj jTX (5)

The first technological constraint (equation 4) incorporates in the model the existence of

minimum time allocations for each activity i, represented by min

iT (min

iT can be zero for certain

activities). The second technological constraint (equation 5) represents the minimum

consumption of a good that is needed when a certain activity is undertaken, with ij representing

the amount of good j needed to perform activity i (per unit time). The Lagrange multipliers

i and j represent the marginal utility of reducing the minimum time for activity i and

reducing the minimum consumption of good j, respectively. Note that this type of relation is like

an aggregated generalization of the implicit set of constraints in Evans’ model (1972) that turns

time use into goods consumption through a matrix Q (see Jara-Díaz, 2003 for a detailed

discussion).

The First Order Conditions (F.O.C.) with respect to the decision variables iT , wT and iX

may be derived in a straightforward fashion as shown in Appendix A. These conditions are:

0 j

ijji

i

i

T

U

[for decision variable Ti] (6)

0w

w

Uw

T

[for decision variable Tw] (7)

.0 jj

j

jP

X

U

[for decision variable Xj] (8)

The F.O.C.’s above have an intuitive interpretation. According to equation (6), activities that are

assigned more than the minimum time necessary ( 0i ) and do not impose a minimum level

9

of consumption on any of the goods, have the same marginal utility, following a common result

in time use models since DeSerpa (1971), who was the first one to propose that all the freely

chosen activities (activities that are assigned more time than necessary, activities that DeSerpa

called “leisure activities”) have the same marginal utility. Of course, the special case is work (see

equation (7)): the marginal utility of time assigned to work plus the wage rate- which is

multiplied by the marginal utility of money- has to be equal to the marginal utility of the

activities that are assigned more than the minimum necessary. In other words, the total value of

work has to be equal to the value of leisure time, as defined by DeSerpa (1971). For activities

that are assigned more than the minimum necessary ( 0i ) and do not impose minimum

consumption for any of the goods, the marginal utility of the time assigned to the activity plus

the marginal utility of a marginal relaxation of the minimum constraint has to be equal to the

marginal utility of the freely chosen activities, as can be seen in equation (6). In the particular

case that one of the activities impose certain minimum good consumption, an extra term has to

be added in the equilibrium: j

ijj . This additional term represents the impact on utility of the

change on the consumption structure when the time assigned to the specific activity is marginally

increased. Finally, according to equation (8), for those goods with a level of consumption greater

than the minimum necessary ( 0j ), the price-normalized marginal utility of good has to be

equal to the marginal utility of money. For those goods that are consumed only the minimum

necessary, the price-normalized marginal utility of good plus the marginal utility of a relaxation

of the minimum consumption constraint has to be equal to the marginal utility of money.

The F.O.C. with respect to kX (equation 8) for FGk (i.e. 0k ) is:

.0 k

k

k PX

U

(9)

Adding (9) overFG and defining

FGk

k we get:

.

FGk

kk XPU

(10)

10

imposing the budget constraint, we can rewrite the denominator of the right side of equation (10)

and get:

RGj

jjfw XPcIwTU

. (11)

Recalling that for RGj ,

i

iijj TX and noting that if we define

RGj

ijji P then we can

write

RGj i

iijj TXP . Given that the summation in the denominator of the right side of

equation (11) can be split into two parts, we can write the following:

.

RFR Ai

ii

Ah

hhfj

Gj

jf TTcIXPcI (12)

As for RAi min

ii TT , there are three terms in the right side of equation (12) that are fixed.

Recalling that the sum of these three terms is defined by Jara-Díaz et al. (2008) as committed

expenses cE , then equation (11) can be re-written as:

.

FAh

hhcw TEwTU

(13)

Dividing equation (9) by U and replacing (13) we obtain the first equation in our system for

FGk :

FAh

hhcwk

kk TEwTXP

. (14)

Now consider the F.O.C. for hT (equation 6) with FAh (i.e. 0h ), which is:

0 j

hjj

h

h

T

U

. (15)

Adding equation (15) over FA and defining

FAh

h we get:

11

F

F

Ah

h

Ah j

hhjj

T

U

T

U

. (16)

We can solve equation (8) for j :

jjjj XUP (17)

Replacing (17) and (13) in (16):

F

F

F

F

Ah

h

Ah j jj

hjhjj

Ah

hhcw

Ah

hh

T

XP

TP

TEwT

T

U

. (18)

Rewriting the denominator of equation (18) based on the total time constraint and defining

committed time as

RAi

ic TT :

cw

Ah j jj

hjhjj

Ah

hhcw

Ah

hh

TT

XP

TP

TEwT

T

U

F

F

F

. (19)

Dividing equation (15) by U and replacing (19) we obtain the second equation in our system for

FAi :

0

j jj

ijijj

Ah

hhcw

ii

cw

Ah j jj

hjhjj

Ah

hhcw

Ah

hh

i

i

XP

TP

TEwT

T

TT

XP

TP

TEwT

T

T

F

F

F

F

(20)

Dividing (7) by U and replacing (13) and (19) we get the third equation of our system:

12

.0

cw

Ah j jj

hjhjj

Ah

hhcw

Ah

hh

Ai

iicw

w

w

TT

XP

TP

TEwT

T

TEwT

w

T

F

F

F

F

(21)

Equations (14), (20) and (21) form a system of 1|||| FF GA equations with the same

number of unknowns. These unknown decision variables are work time, time assigned to those

activities that do not stick to the exogenous minimum, and amount of goods consumed above the

corresponding minimum.

Once the system is solved, the rest of the variables (goods and time) can be found as:

R

ii AiTT min (22)

R

i

iijj GjTX (23)

The value of time as a resource, or value of leisure, can be obtained as:

,

cw

Ai

iicw

Ah j jj

hjhjj

Ah

hhcw

Ah

hh

TT

TEwTXP

TP

TEwT

T

FF

F

F

(24)

and then the value of work can be obtained from equation (7):

.wT

Uw

(25)

2.3 Model with exogenous minimum consumption of goods and endogenous minimum time

allocations

As an alternative model, in addition to (1)-(3) and instead of (4) and (5), we propose the

inclusion of the following family of constraints:

)(0 i

j

jiji iXT (26)

)(0min

jjj jXX (27)

13

The first technological constraint (equation 26) represents the existence of minimum time

allocations that are needed when a certain good is consumed, with ij representing the amount

of time needed to be invested in activity i per unit of consumption of good j. The second

technological constraint (equation 27) incorporates in the model the existence of minimum

consumption of goods for each good j, represented by min

jX ( min

jX can be zero for certain goods).

The First Order Conditions (F.O.C.) with respect to the decision variables iT , wT and iX

may be derived in a straightforward fashion as shown in Appendix B. These conditions are:

0 i

i

i

T

U

[for decision variable Ti ] (28)

0w

w

Uw

T

[for decision variable Tw] (29)

.0 i

iijjj

j

jP

X

U

[for decision variable Xj] (30)

The F.O.C. with respect to hT (equation 28) for FAh (i.e. 0h ) is:

.0

h

h

T

U (31)

Adding (31) overFA and recalling that

FAh

h we get:

.

FAh

hTU

(32)

Imposing the total time constraint, we can rewrite the denominator of the right side of equation

(32) and get:

RAi

iw TTU

. (33)

14

Recalling that for RAi ,

j

jiji XT and noting that if we define

RAi j

ij

jP

then we can

write

j Ai

ijjjR

TXP . Given that the summation in the denominator of the right side of

equation (33) can be split into two parts, we can write the following:

.min

RFR Gj

jjj

Gk

kkk

Ai

i XPXPT (34)

Defining the second term as committed time cT , then equation (33) can be re-written as:

.

c

Gk

kkkw TXPTU

F

(35)

Dividing equation (31) by U and replacing (35) we obtain the first equation in our system for

FAh :

c

Gk

kkkwh

h TXPTTF

. (36)

Now consider the F.O.C. for kX (equation 30) with FGk (i.e. 0k ), which is:

0 i k

iki

kk

k

PXP

U

. (37)

Adding equation (37) over FG and defining

FGk

k we get:

F

F

Gk

kk

Gk i

kiki

XP

U

X

U

. (38)

We can solve equation (28) for i :

iii TU (39)

Replacing (39) and (35) in (38):

15

F

F

F

F

Gk

kk

Gk i

kk

k

ik

i

i

c

Gk

kkkw

Gk

kkk

XP

XPPT

TXPT

XP

U

. (40)

Rewriting the denominator of equation (40) based on the total budget constraint and recalling the

definition of committed expenses, cE :

cw

Gk i

kk

k

ik

i

i

c

Gk

kkkw

Gk

kkk

EwT

XPPT

TXPT

XP

U

F

F

F

. (41)

Dividing equation (37) by U and replacing (41) we obtain the second equation in our system for

FGj :

0

i

jj

j

ij

i

i

cw

j

cw

Gk i

kk

k

ik

i

i

c

Gk

kkkw

Gk

kkk

jj

jXP

PTEwTEwT

XPPT

TXPT

XP

XP

F

F

F

(42)

Dividing (29) by U and replacing (35) and (42) we get the third equation of our system:

.0

c

Gk

kkkw

cw

Gk i

kk

k

ik

i

i

c

Gk

kkkw

Gk

kkk

w

w

TXPT

wEwT

XPPT

TXPT

XP

T

F

F

F

F

(43)

Equations (36), (42) and (43) form a system of 1|||| FF GA equations with the same

number of unknowns. These unknown decision variables are work time, time assigned to those

activities that do not stick to the exogenous minimum, and amount of goods consumed above the

corresponding minimum.

16

Once the system is solved, the rest of the variables (goods and time) can be found as:

R

j

j

iji AiXT min (44)

R

jj GjXX min (45)

The value of time as a resource, or value of leisure, can be obtained as:

,

c

Gk

kkkw

Gk i

kk

k

ik

i

i

c

Gk

kkkw

Gk

kkk

cw

TXPTXPPT

TXPT

XP

EwT

FF

F

F

(46)

and then the value of work can be obtained from equation (7):

.wT

Uw

(47)

3. MODEL ESTIMATION

Considering stochastic error terms ( 1u , iu and jv ) on each F.O.C equation in the model with

endogenous consumption of goods and exogenous time allocations, we have1:

1

~~

1

~~

uTT

XP

TP

TEwT

T

TEwT

w

T cw

Ah j jj

hjhjj

Ah

hhcw

Ah

hh

Ah

hhcw

w

w

F

F

F

F

(48)

1In this view, utility is considered deterministic and stochasticity is introduced in the F.O.C conditions. According to

this view, not only is the consumer aware of all factors relevant to utility formation, but the analyst observes all of

these factors too. However, consumers are assumed to make random mistakes (“errors”) in maximizing utility

(subject to the many constraints), which gets manifested in the form of stochasticity in the F.O.C conditions. Bhat et

al. 2015, in a different context, label such a paradigm as the deterministic utility-random maximization or DU-RM

decision postulate. Earlier, Wales and Woodland (1988) also identified this alternative perspective for utility-based

models – see footnote 5 in their paper, page 268. As discussed in these earlier works, it can certainly be argued that

the DU-RM mechanism is as plausible as the alternative random utility-deterministic maximization (RU-DM)

mechanism used more traditionally in microeconomics. Besides, in the current case, the DU-RM mechanism is

much easier to work with from a practical viewpoint relative to the RU-DM mechanism.

17

F

i

j jj

ijijj

Ah

hhcw

i

cw

Ah j jj

hjhjj

Ah

hhcw

Ah

hh

i

i AiuXP

TP

TEwTTT

XP

TP

TEwT

T

TF

F

F

F

~~

~~

1~

(49)

.~~ F

j

Ah

hhcwjjj GjvTEwTXPF

(50)

where

w

w ~

,

i

i ~

,

~ and

j

j ~ .

In the case of the model with exogenous minimum consumption of goods and endogenous time

allocations, we have:

1

1

~~

~

u

TXPT

wEwT

XPPT

TXPT

XP

Tc

Gk

kkkw

cw

Gk i

kk

k

ik

i

i

c

Gj

jjjw

Gk

kkk

w

w

F

F

F

F

(51)

F

i

Gk

kkkwii AiuXPTTF

~

(52)

.

~~

~~

~F

j

i

jj

j

ij

i

i

cw

j

cw

Gk i

kk

k

ik

i

i

c

Gk

kkkw

Gk

kkk

jj

jGjvXP

PTEwTEwT

XPPT

TXPT

XP

XP

F

F

F

(53)

Due to the existence of the total time constraint (equation 3), only 1L time assignment

equations can be estimated, where L is the number of unconstrained activities. Due to the

existence of the total budget constraint (equation 2), only 1M goods consumption equations

can be estimated, where M is the number of unconstrained goods.

For convenience, we define two new indexes l and m, with 1l referring to work,

Ll ,,3,2 referring to activities corresponding in set

FA ( L is the cardinality of set FA ),

and 1,1,2, Mm referring to goods in set FG ( M is the cardinality of set

FG ). The left

18

hand of each equation is a function of time assigned to activities, ),,( mlww XTTf , ),,( mlwl XTTf ,

and consumed goods ),,( mlwm XTTg . Then, equations (48) to (50) (for the model with

endogenous consumption of goods and exogenous time allocations) or (51) to (53) (for the model

with exogenous minimum consumption of goods and endogenous time allocations), can be

summarized in:

LluXXTTf lMLl ,,2,1,,,,, 111 (54)

1,,2,1,,,,, 111 MmuXXTTg mMLm . (55)

Vector ),,,,,,,(u 12121 ML vvvuuu is then assumed to be a realization from a multivariate

normal distribution, so that ),0(~u 1 ΩMLMVN indicates an )1( ML -variate normal

distribution with mean vector of 0 and covariance matrix .Ω The probability distribution

function of u is denoted by ),0;(.*

1 ΩML . Then the probability that the individual assigns 1T to

work , LTT ,,2 to activities in FA , and 11 ,, MXX to goods in

FG corresponds to:

,,,,,,,,)det(,...,,,,, 12121

*

11121 MLMLML gggfffXXTTTP J (56)

where J is the Jacobian of the vector function

),...,,,,,( 1121 ML XXTTT H ),,,,,,,( 12121

ML gggfff (see Appendix C for the model

with endogenous consumption of goods and exogenous time allocations and Appendix D for the

model with exogenous minimum consumption of goods and endogenous time allocations). Let

ω be the diagonal matrix of standard deviations rω of Ω , and let );(.1 ΔML be the multivariate

standard normal probability distribution function of dimension L+M-1 and correlation matrix Δ .

Then,

1-1-1-ΩωωωJ ;)det(,...,,,,, 1

11

1

1121 H

ML

ML

r

rML XXTTTP (57)

yields the likelihood function:

,,...,,,,,),~,...,~,~,~

,,~

( 1121111 MLML XXTTTPL Ω (58)

where w~~

1 , and Ω is the row vectorization of the upper diagonal elements of Ω . Due to

identification issues, one of the standard deviations of Ω has to be fixed to 1. To ensure that the

19

normalized utility parameters L

~,...,

~2

, ~ and 11~,...,~

M are positive, we parameterize them as

using an exponential function.

4. DATA

4.1 Data description and sample selection

The data used for the analysis is drawn from the LISS panel data. The LISS panel is a

representative sample of Dutch individuals who participate in monthly Internet surveys

(households that could not otherwise participate are provided with a computer and Internet

connection). The panel is based on a true probability sample of households drawn from the

population register, and its first wave was conducted on 2008. A longitudinal survey is fielded in

the panel every year, covering a large variety of domains including work, education, income,

housing, time use, political views, values and personality.

The LISS panel also includes questionnaires designed by researchers with the purpose of

identifying specific behavioral preferences. One of these studies corresponds to a survey on time

use and consumption (see Cherchye et al., 2012 for a detailed description). The first wave of

these questionnaires was implemented in September 2009, a second wave was conducted in

September 2010, and a third in October 2012. In this study, we will focus on the latest wave. The

number of individuals available for the analysis is 5,4632. Respondents reported (1) the time

allocated to 13 activities (including work) during the seven days before the survey, and (2) the

average monthly expenditure (in euros) in 30 categories, considering as reference the past 12

months. The time use and consumption data are complemented with socio-demographic

information drawn from the LISS panel.

The sample used to estimate our model considered individuals who worked at least one

hour during the survey week and who reported expenditure in at least one of the expenditure

categories. Further, we selected workers who live in one-worker households (i.e., the respondent

is the only worker in the household). This last criterion allows assigning all personal and

household expenditures to the sole worker in the household, without making assumptions

regarding how the household expenditures are shared among income producers. Because time

2 The third wave of surveys on time use and consumption was administered to 6,874 households. Out of these

households, 20.5% did not answer the survey and 2.3% returned incomplete surveys.

20

allocation is reported on a weekly basis, the timeframe of our study is a week (including

weekends). Monthly expenditures and monthly income are divided by four to obtain weekly

expenditures and weekly income, respectively.

The database includes the worker’s monthly average gross and net income. For the

analysis, only net income is considered. Henceforth, net income is referred as income. Income is

disaggregated into salary ( wwT ) and non-work income ( I ): salary is obtained from working (for

an employer or independently) and it is used to compute the wage rate, while non-work income

corresponds to the earnings received from pensions, investments, annuities, governmental

support, scholarships, tax reimbursement and others non-work related sources.

Several consistency checks were performed to obtain the estimation sample. First,

workers with relevant but missing data (such as income and time allocation) were removed from

the sample. Second, workers who reported sleeping on average less than 4 hours per day were

also removed from the sample (accounting for 2.5% of the workers). We hypothesize that

individuals who reported sleeping less than 28 hours per week may have underestimated their

sleeping time and, therefore, misestimated the time assigned to other activities. Third, we

removed from the sample those workers who reported extremely high activity durations (for

example, some people reported working 168 hours per week). Fourth, respondents who spent

less than 2 euros per week were removed from the sample, along with those workers whose wage

was less than 3 euros/hour (the minimum hourly wage in Netherlands was about 8.4 euros in

2012). Finally, we noticed that some workers´ expenditure was higher than their income. To

correct this inconsistency, we removed from the sample those observations where the difference

between expenditure and income was greater than 20% of the worker´s income. If the difference

between expenditures and income was smaller than 20% of the worker´s income, the difference

was added to the worker´s non-work income )(I . Therefore, in these last cases, the difference

between expenditures and income is zero. After this selection process, the estimation sample

included 1,193 workers.

4.2 Classification of activities and association of expenditures

The 13 activities available in the original database were grouped into the following 11 activities

(three activities –helping parents, helping family members and helping non-family members–

were combined into one –assisting friend and family– due to low participation):

21

1) Work: any type of paid work as an employee or as a self-employed worker. The reported

time includes overtime hours.

2) Commute: travel to and from work, including trips to intermediate stops (such as passing

by shops or markets during the way back home).

3) Household chores: cleaning, shopping, cooking, gardening, etc.

4) Personal care: washing, dressing, eating, visiting the hairdresser, seeing the doctor, etc.

5) Education: includes day or evening courses, professional courses, language courses or other

course types, doing homework, etc.

6) Activities with children: any activity with own children aged less than 16 years, such as

washing, dressing, playing, taking child to see doctor, taking child to school/hobby

activities, etc.

7) Entertainment: in-home and out-of-home recreational activities, such as watching TV,

reading, practicing sports, hobbies, computer as hobby, visiting family or friends, going

out, walking the dog, cycling, sex, etc.

8) Assisting friends and family: assistance to friends and family members (not children). For

example: helping with administrative chores, washing, dressing, seeing the doctor,

voluntary work, babysitting, etc.

9) Administrative chores and family finances.

10) Sleeping and relaxing: sleeping, resting, thinking, meditating, being ill, etc.

11) Going to church and other activities: going to church, attend funeral/wedding, and any

activity not considered above.

To incorporate the novel set of constraints in our model in a way that allows us to make an

easy interpretation of the results, we need the expenditure corresponding to the goods allocated

to each activity purpose during the survey week. In this way, we are able to relate to each non-

work activity n an associated time ( nT ) and an associated expense ( nn XP ), the latter being the

money expenditure related to a composite good nX that includes all the goods necessary to

perform activity n. This one-by-one relation between time allocation and good consumption has

been a common assumption in the time use microeconomic framework (see for example Becker,

1965, Chiswick, 1967, De Serpa, 1971, Evans, 1972, De Donnea, 1972, Bruzelius, 1979, Juster,

1990, Jara-Díaz, 2003, Jara-Díaz et al., 2008, Konduri et al., 2011, and Jara-Díaz and Astroza,

2013), either for theoretical reasons or because data limitations. In our case, our model

22

formulation does not require this assumption and the LISS data offers a broad range of

possibilities regarding time use and good consumption structures, but certainly this assumption

makes the interpretation of the results easier.

The LISS panel database contains detailed information about expenditure in 30 distinct

categories, but these categories do not directly relate to the activity purposes listed above.

Consequently, we needed to associate the expenditures to activities. For this purpose, the

expenditure categories were studied in detail to identify those that matched the description of the

activities. In addition, we computed the fixed expenditures )( fc as those expenditures not

related with any activity purpose. Details about the association procedure and the definition of fc

can be found in Appendix E. Descriptive statistics are presented in Table 1.

Table 1: Descriptive statistics

Activity Participation

(%)

Duration (hours/week)* Expenditure (euros/week)

*

Mean St.

Dev. Min. Max. Mean

St.

Dev. Min. Max.

Work 100.0 33.4 13.7 1.0 100.0 - - - -

Commute 94.0 4.8 4.8 0.2 60.0 12.8 14.2 0.0 216.0

Household chores 97.8 12.4 9.8 0.3 90.0 5.9 9.8 0.0 107.5

Personal care 100.0 9.1 5.8 0.5 49.0 96.9 66.5 0.0 1,005.0

Education 24.7 7.4 9.3 0.2 87.7 1.4 7.4 0.0 125.0

Activities with children 31.2 14.3 11.7 0.5 65.0 17.6 29.1 0.0 166.3

Entertainment 99.8 31.9 16.1 1.0 102.0 38.7 63.1 0.0 725.0

Assisting friends and family 57.6 7.5 7.8 0.2 81.3 - - - -

Administrative chores and

family finances 86.6 3.1 3.5 0.2 50.0 - - - -

Sleeping and relaxing 100.0 58.8 11.4 28.0 119.2 - - - -

Going to church and other

activities 42.5 11.7 12.5 0.3 71.0 - - - -

Fixed expenditures fc 92.5 - - - - 330.7 179.8 2.4 1,316.0

Number of observations 1,193

(*): Durations and expenditures are computed only for workers participating in the corresponding activity.

By construction, all individuals in the sample allocate time to work and sleeping/relaxing

activities: on average, individuals work 6.6 hours per weekday and sleep/relax 8.4 hours per day.

In addition, all workers spend time in personal care activities (recall that this activity type

includes eating and dressing). Most workers allocate some time to commute, entertainment and

personal care, while education and activities with children present the lowest participation rates.

23

Regarding expenditure, personal care presents the highest value and it is also the most

expenditure-intensive activity (average of 10 euros/hour). Although people spend a relatively

large amount of money in entertainment activities, these represent only an expenditure rate of 2.2

euros/hour, which is considerably lower than the average wage of 18 euros/hour.

5. MODEL ESTIMATION RESULTS

5.1 Variable specification and model formulation

To estimate the model, the first step is to classify the 10 non-work activities into the sets defined

in our model. This classification is presented in Table 2. Activities with restricted expenses RG

are subdivided into activities with expenditure restricted at its minimum )( R

minG and activities

with expenditure restricted to zero (R

zeroG ); in other words, .R

zero

R

min

R GGG Although this

distinction is irrelevant from a model estimation perspective, we believe that it is important to

develop an accurate activity classification that recognizes the characteristics of the data used for

the analysis.

Table 2: Classification of activities

Sets

Restricted expenses RG

Unrestricted

expenses FG

Restricted at

minimum R

minG

Restricted at

zero R

zeroG

Restricted

activities RA

- Household chores

- Personal care

- Commute

- Education

- Assisting friends and

family

- Administrative chores and

family finances

Unrestricted

activities FA

- Activities with children

- Entertainment

- Sleeping and relaxing

- Going to

church and other

activities

The set of activities than belong to RA (activities restricted in time) and have an

associated expense that belongs to RG (activities restricted in expenses) comprises the

following 6 activities: household chores, personal care, assisting friends and family,

administrative chores and family finances, commute and education. That is, individuals spend

the smallest amount of time performing these activities, as well as stick to the minimum

24

monetary resources needed to perform the activity. Traditionally personal care and

administrative/household chores and personal care have been classified as restricted time

activities (see for example Aas, 1982, Bittman and Wajcman, 2000, and Robinson and Godbey,

2010) because of their maintenance-oriented nature. People need to take care of their health and

hygiene, and manage the household maintenance. These maintenance activities are generally

driven by a physical need, but in most cases, individuals do not want to spend more money than

necessary to perform such activities (Gronau and Hamermesh, 2006 classified maintenance

activities as goods intensive, i.e., individuals really care about the amount of goods they are

spending in order to perform these activities; the reader is also referred to Ahn et al., 2005 who

observes that individuals generally try to save money in maintenance activities). Similarly, there

are other tasks- such as assisting friends and family or family finances- that have to be taken care

of. Regarding commute activity, we believe that individuals will assign the minimum necessary

because, in general, individuals would rather be doing something else, either at home, at work, or

somewhere else, than riding a bus or driving a car. So they will assign the minimum necessary

time to commute and, of course, they will not spend more money than necessary no matter which

mode of transportation they choose (see Mokhtarian and Chen, 2004 for a review of different

studies of travel time and related money expenditures). Finally, we consider that individuals will

spend the minimum necessary time in education because classes have a fixed length that usually

individuals cannot choose, assignments are mandatory tasks, and extra time of study does not

mean extra pay. The expenses associated to commute, household chores, personal care, and

education belong to the set RGmin and the expenses associated to assisting friends and family, and

administrative chores and family finances to R

zeroG . There is no activity belonging to RA with an

associated expense belonging to FG (unrestricted regarding expenses, but restricted regarding

time). Activities with children, entertainment, and sleeping and relaxing are considered “time

unrestricted and expenses restricted” activities, i.e. they belong to FA and their associated

expense belongs to RG . Finally, “going to church and other activities” is the only activity in

FA

and with its associated expense belonging to FG (unrestricted in terms of time and expenses).

Due to the model derivation it is required that workers allocate some positive expense to

activities with associated expenses in set FG . However, 57.5% of the sample does not participate

in “going to church and other activities”, and there are no expenses associated with this activity

25

in the data. Then, to estimate the model, a small expense (2 euros/week) was appended to “going

to church and other activities” for each worker.3

As discussed in Section 3, one of the freely chosen activities and one of the freely chosen

goods cannot be estimated. The time assigned to “going to church and other activities” and the

corresponding expenses are not considered as dependent variables. Consequently, there are four

dependent variables in our system (we have a system of four equations): time assigned to work

( wT ), time assigned to activities with children childT( ), time assigned to entertainment ( entT ), and

time assigned to sleeping and relaxing ( sleepT ). There are five utility parameters ( w~

, child~

, ent~

,

sleep~

and )~ and nine covariance matrix elements to be estimated (w is fixed to 1 for

identification). Since we are considering a one-by-one relation between time assigned to

activities and good expenses, we can rewrite the endogenous minimum constraints as nnn TX ,

for the model with endogenous minimum consumption of goods (equation 5), and nnn XT ,

for the model with endogenous minimum time allocations (equation 26). For the model with

exogenous minimum times and endogenous minimum consumption of goods, the terms nnP

were directly computed from the data by dividing time by the associated expenditure. As we

mentioned in section 2, the estimation does not require the value of nP and n independently.

With the values of nnP we can obtain the n values. Similarly, for the model with endogenous

minimum times and exogenous minimum consumption of goods, the terms nn P were directly

computed from the data by dividing expenditure by the associated time. With the values of

nn P we can obtain the n values.

5.2 Estimation results

Tables 3 and 4 present the model estimation results for the two different versions of the model:

with exogenous minimum times and endogenous minimum consumption of goods (Table 3), and

the model with endogenous minimum times and exogenous minimum consumption of goods

(Table 4). The upper section of the tables presents the model parameters and the lower section

3 A sensitivity analysis was performed to investigate the repercussions of doing so. For values of less than 2 euros,

the model could not be estimated. For values equal or greater than 2 euros, the coefficients could be estimated and

there was no substantial difference in the results for values between 2 and 25 euros.

26

shows the average computed values of time. For each version of the model we estimated two

different error structures: one with identically and independently distributed (iid) error terms and

one with a full covariance matrix (four models were estimated in total). A likelihood ratio test

shows that the model incorporating a flexible structure of the error is statistically superior,

validating the procedure proposed in this paper (the likelihood ratio test statistic is 484 for the

with exogenous minimum times and endogenous minimum consumption of goods and 478 for

the model with endogenous minimum times and exogenous minimum consumption of goods,

which is much larger than the table chi-squared value with two degrees of freedom at any

reasonable level of significance). Tables 3 and 4 show that all the parameters are statistically

significant at the 95% level of confidence, but ~ is associated with a p-value of only 0.27 in the

model with full covariance matrix for both technical constraint formulations.

Table 3: Model parameters; exogenous minimum time, endogenous minimum consumption

Model with iid errors

Model with full

covariance matrix

Model parameters

Coefficients Estimate t-stat Estimate t-stat

Utility

w~

0.095 6.10 0.093 5.04

child~

0.017 7.22 0.014 6.17

ent~

0.078 5.89 0.077 5.19

sleep~

0.686 11.81 0.691 10.23

~ 0.1003 3.09 0.1001 1.11

Covariance matrix

childwork 0.000 - 0.1736 7.58

childent 0.000 - 0.1104 4.59

Log-likelihood -11,302.8 -11,060.8

Number of observations 1,193 1,193

Average values of time [euros/hr]

Estimate Std. dev. Estimate Std. dev.

Leisure 59.57 93.35 59.42 93.11

Work 41.57 95.81 41.42 95.58

Wage 17.99 24.03 17.99 24.03

Ratio leisure – wage 3.31 3.30

Ratio work – wage 2.31 2.30

27

Table 4: Model parameters; endogenous minimum time, exogenous minimum consumption

Model with iid errors

Model with full

covariance matrix

Model parameters

Coefficients Estimate t-stat Estimate t-stat

Utility

w~

0.090 6.00 0.091 4.98

child~

0.016 6.67 0.014 6.05

ent~

0.069 5.90 0.079 5.03

sleep~

0.672 11.02 0.689 9.99

~ 0.100 3.10 0.1002 1.10

Covariance matrix

childwork 0.000 - 0.1736 7.58

childent 0.000 - 0.1104 4.59

Log-likelihood -11,290.8 -11,024.2

Number of observations 1,193 1,193

Average values of time [euros/hr]

Estimate Std. dev. Estimate Std. dev.

Leisure 59.30 93.32 59.32 93.10

Work 42.00 95.78 41.03 95.61

Wage 17.99 24.03 17.99 24.03

Ratio leisure - wage 3.31 3.30

Ratio work - wage 2.31 2.30

The value of the parameters do not have a direct interpretation since they are ratios between

exponents of the Cobb-Douglas utility function, but the ratio between each pair of i~

(including

w~

) can be interpreted as the ratio between the elasticities associated with the corresponding

variable. Comparing two coefficients can give an idea of the relative importance of each

activity/good in terms of utility. In the four models, both technical constraint configurations with

both covariance matrix configurations, sleep is the activity with highest impact on utility,

following by work, entertainment and, finally, child-care. When the full covariance matrix

models were estimated, we found positive correlation between the error term associated with the

child-care equation and the error terms associated with the work equation and the entertainment

28

equation. This means that the unobservable factors explaining the child-care equation are also

present in the work equation and the entertainment equation. Regarding the values of time, both

the value of leisure and the value of work are positive for the four models; consequently, workers

considered in the analysis extract pleasure (at the margin) both from working and undertaking

leisure activities. For the model with endogenous minimum consumption of goods and

exogenous time allocations, the value of time of leisure is estimated as 59.6 euros/hour and the

value of time of work as 41.6 euros/hour. For the model with exogenous minimum consumption

of goods and endogenous time allocations the results are slightly different; the value of time of

leisure is estimated as 59.3 euros/hour and the value of time of work as 42.0 euros/hour. If all the

activities belonging to RA would have an associated monetary expenditure belonging to

RG ,

then we can write for those activities the endogenous minimum constraints as nnn TX , for the

model with endogenous minimum consumption of goods, and nnn XT , for the model with

endogenous minimum time allocations. If in addition, all the activities belonging to FA would

have an associated monetary expenditure belonging to FG , i.e. the freely chosen activities/goods

do not impact the technological constraints, then we can simply write the relation nn 1 and

both models (with both technical constraint formulations) would be equivalent. However, in our

specification some of the activities (activities with children, entertainment and sleeping and

relaxing) have associated restricted expenses but are unrestricted about time. This breaks the

symmetry between FA and

FG and provokes differences in the value of the estimated Cobb-

Douglas coefficients and, consequently, differences in terms of the value of time.

As mentioned in the previous sections, the novelty of the proposed model is the

introduction of a link between minimum consumption and time. To assess the contribution of our

approach, we estimated a model that does not incorporate this link, as developed by Jara-Díaz et

al. (2008), and computed the corresponding values of time. The model estimation results can be

found in Appendix F. The resulting values of leisure and work are 122.8 euros/hour and 104.8

euros/hour, respectively. If we compare the value of leisure and work among the models, we can

identify a clear difference: the model without the link overestimates the values of time. In other

words, when omitting the relation between consumption and time, the model cannot correctly

capture the individual valuation of time. Intuitively, since we are considering in one version of

the current model that some of the freely chosen activities (those that individuals assign more

29

time than the minimum necessary) are imposing lower bounds (or minimum requirements) to

goods consumption and- consequently- expenses, leisure time has a “cost” in this formulation.

This differs from previous models- without our link between goods and activities- that only

consider a pure cost-free leisure time. In the other case, when we are considering that some of

the goods are imposing lower bounds to time assigned to those activities that are restricted to the

minimum, then the consumption of goods is directly related to the restricted time and

consequently, to the time available for leisure. Since the model without the link does not

consider this ‘time cost’ of the consumption of goods, the value of leisure time is overestimated.

Finally, we explored several segments of the population to identify differences in the

valuation of time. The segmentation was made using socio-demographic data, including age,

gender, income, education, location of the household (urban vs. rural), presence of children in

the household and whether the worker had a partner. Further, we explored combinations of the

previous segments (for example, we compared the values of time of females with children and

females without children). Table 5 reports those segments that showed statistical differences in

their valuations of time using the model with exogenous minimum times and endogenous

minimum consumption of goods only, as both versions yield very similar results.

Table 5: Values of time for different segments of the population

Presence of children in the household Age

No children At least one child 50 years > 50 years

Estimate Std.dev. Estimate Std.dev. Estimate Std.dev. Estimate Std.dev.

Leisure 69.75 101.83 2.48 3.78 5.74 9.09 93.70 144.51

Work 50.35 103.76 -13.34 17.39 -8.85 15.64 70.61 147.58

Wage 19.40 27.56 15.82 17.00 14.59 13.01 23.09 33.86

Ratio leisure-wage 3.60 0.16 0.39 4.06

Ratio work-wage 2.60 -0.84 -0.61 3.06

Location of household Income level

Urban area Not urban area Low income High income

Estimate Std.dev. Estimate Std.dev. Estimate Std.dev. Estimate Std.dev.

Leisure 75.69 117.42 60.62 95.76 17.82 25.11 158.11 188.30

Work 57.37 120.02 42.86 97.45 1.29 32.69 133.72 192.65

Wage 18.32 28.76 17.76 20.02 16.53 20.60 24.38 34.68

Ratio leisure-wage 4.13 3.41 1.08 6.48

Ratio work-wage 3.13 2.41 0.08 5.48

30

The value of leisure is positive (as expected) for all segments and the value of work is

negative for some the segments as follows. Workers who have children present a negative value

of work time, indicating that they do not extract pleasure from work at the margin; on the other

hand, workers without children enjoy their work at the margin. This result could be related with

the financial freedom perceived by workers who do not have to economically support children:

they can choose a more satisfying job than workers who need to provide for their family.

According to several earlier studies (see, for example, Kim et al., 2005, Uunk et al., 2005, Baxter

et al., 2007, and Compton and Pollak, 2014), parents - especially women - perceive less job

autonomy (the freedom to decide how they do their work) and more pressure regarding job

location selection than individuals without children. Another explanation is that parents prefer to

spend time out of work to share it with their children (Sayer et al., 2004).

Young workers (aged less or equal than 50 years) have a negative value of work, while

older workers (aged more than 50 years) have a positive one. It is possible that young workers,

compared to old workers, have more debt or commitments (college debt, mortgage) that, to some

extent, force them to choose unsatisfying jobs.4 Also, earlier studies have shown that older

workers generally have more positive job attitudes (such as overall job satisfaction, satisfaction

with work itself, satisfaction with pay, job involvement, emotional exhaustion, or satisfaction

with coworkers) than younger workers (see Rhodes, 1983, Carstensen, 1992, Mather and

Johnson, 2000, and Ng and Feldman, 2010). Regarding the location of the household, Table 4

shows that the valuation of time is higher for workers living in urban areas (although wages are

statistically the same). A plausible explanation of this result is that workers in urban areas can

participate in many activities that are not feasible in rural areas, such as attending cultural

activities, shopping and eating out. Then, due to increased accessibility, these workers perceive

their times as more valuable than workers whose houses are located in rural areas (Farrington

and Farrington, 2005). Finally, income is a relevant determinant of value of time. Our results

show that low income workers (monthly income less or equal to 4,000 euros) have a lower

valuation of time than high income workers (monthly income greater than 4,000 euros).

4 We hypothesized that young workers were more likely to have children living with them than old workers. A

hypothesis test concluded that there is no significant correlation between age and presence of children in the

household, rejecting our initial hypothesis.

31

6. CONCLUSIONS

We have developed a model explicitly introducing a piece that was missing in previous models

of time use, namely a relation between goods consumed and the time assigned to activities that

use it. Although a closed solution for activities and work time could not be found, this indeed

improves over the previous most advanced microeconomic formulations because minimum

levels of consumption or time assignment to activities become endogenous. However, we have

generated a system of equations where the decision variables are work, those activities that are

assigned more than the minimum and those goods that are consumed more than needed. From

this system, the parameters of the implicit equations can be estimated using maximum likelihood

techniques without assuming independence of the error terms. We further discuss identifiability

issues and explicitly compute the Jacobian resulting from the likelihood multivariate integral.

Our microeconomic framework is applied to a Dutch weekly time use and consumption

database. To our knowledge, this is one the few surveys in the world that includes both time

allocation and good consumption information. Using the estimated model parameters, we

computed the values of time (value of leisure and value of work), which are considerable higher

than the wage. A comparison of these estimates with those from a model that does not include

the additional constraint shows substantial differences: the values of time for the model without

the link are about twice the values of time of our proposed model, showing the importance of

correctly introducing relations between time allocation and good consumption in the modeling

framework. This empirical result is particularly relevant from a policy standpoint, as a

miscalculation of the value of time can lead to erroneous computation of the benefits of public

investment projects. In addition, value of time estimations were performed on different segments

of the population. Significant differences in the valuation of time were observed when

segmenting by income, age, location of the household (urban vs. rural) and presence of children

in the household, providing interesting insights regarding Dutch workers’ preferences and

lifestyles.

But a better understanding of the social elements behind the perception, valuation and use

of time is not the only practical use of the improved models. Forecasting changes in time use

after changes in technology as faster transit services or improved ways to do errands (e.g.

teleshopping), is also feasible with the estimated models because the equations systems could be

used to simulate the impact by simply varying the (exogenous) parameters affected: min goods

32

or min times levels, or the alpha-price. So, beyond a better understanding of the impact of

technical change on consumption and time use at a micro level, knowing the new time

assignments and the value gained by the individuals through the extra leisure is of great

importance from a practical viewpoint.

By way of future extensions, we are working on a discrete choice framework for the

decision to assign time to activities. Participation choice could be very important to address the

censured nature of time allocation, and could allow a non-arbitrary mechanism for observations

with zero values for the dependent variables. This additional discrete dimension in our model

would certainly be interesting for future research.

ACKNOWLEDGEMENTS

We thank the Institute for Complex Engineering Systems (grants ICM: P-05-004-F and

CONICYT: FBO16), Fondecyt grant 1160410, the TUO network and the ACTUM project for

partial funding of this research. We also thank the CentERdata (Tilburg University, The

Netherlands) who, through the MESS project funded by the Netherlands Organization for

Scientific Research, collected the LISS panel data used in this analysis. Finally, we thank

Frederic Vermeulen and Arthur van Soest for insights and help regarding this data set.

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36

APPENDIX A: DERIVATION OF FIRST ORDER CONDITIONS FOR THE MODEL WITH EXOGENOUS

MINIMUM TIMES AND ENDOGENOUS MINIMUM CONSUMPTION OF GOODS

The Lagrangian function is given by:

j i

iw

i

fiiwj

i

iw TTcXPIwTXTTL jiw )()(),(

XT (A.1)

j i

iijjj

i

iii TXTT )()( min

Then the partial derivative of the lagrangian respect to each decision variable is:

j

ijji

i

i

i T

U

T

L

(A.2)

w

T

U

T

L

w

w

w

(A.3)

.jj

j

j

j

PX

U

X

L

(A.4)

APPENDIX B: DERIVATION OF FIRST ORDER CONDITIONS FOR THE MODEL WITH

ENDOGENOUS MINIMUM TIMES AND EXOGENOUS MINIMUM CONSUMPTION OF GOODS

The Lagrangian function is given by:

j i

iw

i

fiiwj

i

iw TTcXPIwTXTTL jiw )()(),(

XT (B.1)

j

jjj

i j

jijii XXXT )()( min

Then the partial derivative of the Lagrangian respect to each decision variable is:

i

i

i

i T

U

T

L (B.2)

w

T

U

T

L

w

w

w

(B.3)

.

i

jijij

j

j

j

PX

U

X

L

(B.4)

37

APPENDIX C: COMPUTATION OF THE ELEMENTS OF THE JACOBIAN – ENDOGENOUS

MINIMUM CONSUMPTION OF GOODS AND EXOGENOUS MINIMUM TIMES

The elements of the Jacobian are given by:

11for,,,,,

2for,,,,,

1for,,,,,

111

111

111

MLhLX

XXTTf

LhT

XXTTf

hT

XXTTf

J

h

MLl

h

MLl

w

MLl

lh

(C.1)

where function lf is defined in equation (21) for 1l (work), in equation (20) for Ll ,,3,2

(activities in FA ), and (14) for 1,,1 MLLl (activities in )FG . Let

~1

2

FAi

iicw TEwTC (C.2)

2

1

cw TTD

(C.3)

Then, the lhth

element of the Jacobian is:

DCwTJ ww 22

11

~ (C.4)

11for0

2for1

MLhL

LhDwCPJ

hh

h

(C.5)

12for

12for

1

1

1LhLD

LhCwPDJ

ll

l

(C.6)

1,2for

12and22for

12and12for0

1,2for

12

11

11

12

LhlLzT

LhLlLD

LhLLl

LhlPCPDzT

J

lh

l

l

llhhlh

l

l

lh

~

~

(C.7)

where 1lhz if l h and 0lhz if hl . There is no closed-form structure for the determinant

of the Jacobian.

38

APPENDIX D: COMPUTATION OF THE ELEMENTS OF THE JACOBIAN – EXOGENOUS MINIMUM

CONSUMPTION OF GOODS AND ENDOGENOUS MINIMUM TIMES

The elements of the Jacobian are given by:

11for,,,,,

2for,,,,,

1for,,,,,

111

111

111

MLhLX

XXTTf

LhT

XXTTf

hT

XXTTf

J

h

MLl

h

MLl

w

MLl

lh

(D.1)

where function lf is defined in equation (29) for 1l (work), in equation (30) for

1,,3,2 1 Ll (activities in FA ), and (31) for 1,,21 LLl (activities in )FG . Let

~1

2

FAi

iicw TEwTC (D.2)

2

1

cw TTD

(D.3)

Then, the lhth

element of the Jacobian is:

DCwTJ ww 22

11

~ (D.4)

12for0

12for

1

1

1LhL

LhDwCPJ

hh

h

(D.5)

12for

12for

1

1

1LhLD

LhCwPDJ

ll

l

(D.6)

1,2for

12and22for

12and12for0

1,2for

12

11

11

12

LhlLzT

LhLlLD

LhLLl

LhlPCPDzT

J

lh

l

l

llhhlh

l

l

lh

~

~

(D.7)

where 1lhz if l h and 0lhz if hl . There is no closed-form structure for the determinant

of the Jacobian.

39

APPENDIX E: ASSOCIATION OF EXPENDITURES TO ACTIVITIES

Activity Expenditure category considered

Commute

Average weekly household expenditure transportation, multiplied by 0.36.

Assumption: According to a recent study in the Netherlands, about 18% of all trips

are trips to work (Bohte and K. Maat, 2009). Then, trips to and from work account

for about 36% of all trips.

Household

chores

Average weekly household expenditure in cleaning the house or maintaining the

garden, divided by the number of adults in the household.

(Assumption: all adults in the household equally participate in household chores).

Personal care

Average weekly personal expenditure in eating at home.

Assumption: Some respondents did not declare their expenditure in this category,

but they reported the household expenditure in eating at home (for all household

members). Then, for those respondents with missing data, this expenditure was

computed as the household expenditure divided by the household size. The

assumption is that all household members consume the same amount of food. This

was validated by computing the proportion of personal expenditure in eating at

home, compared to the total household expenditure which, in average, was

consistent with this assumption.

Average weekly personal expenditure in food and drinks outside the house.

Average weekly personal expenditure in personal care products and services.

Average weekly personal expenditure in medical care and health costs not covered

by insurance.

Education Average weekly personal expenditure in (further) schooling.

Activities with

children

Expenditure per week for children living at home in: food and drinks outside the

house, cigarettes and other tobacco products, clothing, personal care products and

services, medical care and health costs not covered by insurance, leisure time

expenditure, (further) schooling, donations and gifts, other expenditures.

Assumption: all children-related expenditure is associated with the time spent with

them. This is not necessarily true: a parent can buy food and drinks outside the

house for the children, but not spend time while the children eat. Or he/she can

purchase a movie ticket and do not go with the children to the cinema. However,

the survey does not provide information that allows us to identify the relationship

between expenditures and activities with children, and we decided to consider all

expenditures related with children.

Entertainment

Average weekly personal expenditure in leisure time expenditure.

Average weekly household expenditure in daytrips and holidays with the whole

family or part of the family.

Assisting

friends and

family

No expenditure.

Administrative

chores and

family

finances

No expenditure.

Sleeping and

relaxing No expenditure.

Going to

church and

other activities

No associated expenditure.

40

In addition, we constructed the fixed expenses fc considering the following expenditure

categories:

Average weekly household expenditure in mortgage and rent.

Average weekly household expenditure in general utilities and insurances.

Average weekly household expenditure in children’s daycare.

Average weekly household expenditure in alimony and financial support for children not

(or no longer) living at home.

Average weekly household expenditure in debts and loans.

Average weekly household expenditure in other household expenditure.

Average weekly household expenditure transportation, multiplied by 0.64 (corresponds to

the expenditure on transportation for other household members and non-work travel).

Average weekly household expenditure in eating at home, minus average weekly

household expenditure in eating at home (corresponds to the expenditure in eating at-

home for other household members).

41

APPENDIX F: MODEL WITH NO LINK BETWEEN CONSUMPTION AND TIME ALLOCATION

Coefficient Model with iid errors

Model with full

covariance matrix

Estimate t-stat Estimate t-stat

Utility

0.489 44.82 0.488 366.69

0.105 7.33 0.104 66.86

child 0.036 1.57 0.036 16.98

ent 0.254 10.93 0.255 73.70

sleep 0.460 18.73 0.461 162.31

other 0.040 1.77 0.040 17.90

Covariance matrix

work 100.000 - 106.101 24.42

childwork 0.000 - 9.824 3.53

entwork 100.000 - 85.707 24.42

entwork 0.000 - 53.234 11.28

otherwork 0.000 - -47.331 -11.18

child 100.000 - 223.192 24.42

entchild 0.000 - 31.106 8.83

sleepchild 0.000 - -19.435 -6.24

otherchild 0.000 - -62.645 -11.90

ent 100.000 - 130.307 24.40

sleepent 0.000 - 0.000 -

otherent 0.000 - 0.000 -

sleep 0.000 - 0.000 -

othersleep 0.000 - 0.000 -

other 100.000 - 98.168 4.02

Log-likelihood -32,989.7 -22,152.3

Number of observations 1193 1193

Estimate Std. dev. Estimate Std. dev.

Leisure 132.67 234.16 122.80 215.23

Work 114.68 213.00 104.81 194.08

Wage 17.99 24.03 17.99 24.03